The exactly integrable systems connected with semisimple algebras of the second rank $A_2,B_2,C_2,G_2$

TL;DR

This paper explicitly presents all exactly integrable systems related to second-rank semisimple algebras, providing their general solutions via fundamental group representations, advancing understanding of algebraic integrability.

Contribution

It offers a comprehensive explicit classification of integrable systems linked to second-rank semisimple algebras with solutions expressed through fundamental representations.

Findings

Complete explicit forms of integrable systems for second-rank semisimple algebras.

General solutions expressed in terms of matrix elements of fundamental representations.

Unified framework for integrable systems connected with various second-rank algebras.

Abstract

All exactly integrable systems connected with the semisimple algebras of the second rank with an arbitrary choice of the grading in them are presented in explicit form. General solution of such systems are expressed in terms of the matrix elements of two fundamental representations of corresponding semisimple groups.